To determine Troy Juth's monthly payment for a loan of [tex]$73,000 at an annual interest rate of 8.5% over 36 months, we need to follow several steps. The critical part of this process involves using the formula for calculating the monthly payment for an installment loan. Here are the steps:
1. Convert the Annual Interest Rate to a Monthly Interest Rate:
- Annual interest rate = 8.5%
- Monthly interest rate = (8.5 / 12) / 100 = 0.85 / 12 = 0.00708333
2. Set Up the Monthly Payment Formula:
- The formula for calculating the monthly payment (P) is:
\[
P = \frac{r \times PV}{1 - (1 + r)^{-n}}
\]
- Where:
- \( r \) = monthly interest rate = 0.00708333
- \( PV \) = loan amount = $[/tex]73,000
- [tex]\( n \)[/tex] = loan term in months = 36
3. Calculate the Monthly Payment:
- Plugging in the values:
[tex]\[
P = \frac{0.00708333 \times 73,000}{1 - (1 + 0.00708333)^{-36}}
\][/tex]
4. Simplify the Expression:
- First, calculate [tex]\( (1 + r)^{-n} \)[/tex]:
[tex]\[
(1 + 0.00708333)^{-36} \approx (1.00708333)^{-36} \approx 0.775904
\][/tex]
- Next, calculate [tex]\( 1 - (1 + r)^{-n} \)[/tex]:
[tex]\[
1 - 0.775904 \approx 0.224096
\][/tex]
- Finally, calculate the numerator [tex]\( r \times PV \)[/tex]:
[tex]\[
0.00708333 \times 73,000 \approx 517.083
\][/tex]
- Now divide the numerator by the denominator to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{517.083}{0.224096} \approx 2304.43
\][/tex]
5. Round to the Nearest Cent:
- The calculated monthly payment is approximately 2304.43.
So, Troy Juth's monthly payment for the loan will be $2304.43.
